On a discrete fractional stochastic Grönwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale

Hendy A.S. Zaky M.A. Doha E.H.
1 April 2023 De Gruyter Open Ltd

International Journal of Nonlinear Sciences and Numerical Simulation
2023 #24 Issue 2 531 - 537 pp.

The aim of this paper is to derive a novel discrete form of stochastic fractional Grönwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Grönwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < α < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper.

a priori estimate , discrete stochastic fractional Grönwall inequality , L1 interpolation schemes , martingale

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Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg, 620002, Russian Federation
Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt
Department of Applied Mathematics, National Research Centre, Dokki, Cairo, 12622, Egypt
Department of Mathematics, Nazarbayev University, Nur-Sultan, Kazakhstan
Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

Department of Computational Mathematics and Computer Science
Department of Mathematics
Department of Applied Mathematics
Department of Mathematics
Department of Mathematics

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